This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379734 #10 Jan 07 2025 19:05:18
%S A379734 0,1,1,2,1,1,1,4,3,2,1,8,1,4,8,27,1,32,1,40,24,13,1,175,56,22,188,166,
%T A379734 1,387,1,874,166,61,410,1833,1,98,391,3028,1,2704,1,1828,5893,239,1,
%U A379734 16756,3446,9742,1865,5276,1,32927,8179,31643,3840,814,1,82958,1
%N A379734 Number of integer partitions of n into parts > 1 whose product is a multiple of n.
%C A379734 Allowing 1's gives A057568.
%e A379734 The a(n) partitions for n = 6, 10, 12, 15, 22:
%e A379734 (6) (10) (12) (15) (22)
%e A379734 (5,3,2) (6,6) (6,5,4) (11,6,5)
%e A379734 (5,4,3) (7,5,3) (11,7,4)
%e A379734 (6,4,2) (10,3,2) (11,8,3)
%e A379734 (4,3,3,2) (5,4,3,3) (11,9,2)
%e A379734 (5,3,2,2) (5,5,3,2) (11,4,4,3)
%e A379734 (6,2,2,2) (6,5,2,2) (11,5,4,2)
%e A379734 (3,3,2,2,2) (5,3,3,2,2) (11,6,3,2)
%e A379734 (11,7,2,2)
%e A379734 (11,3,3,3,2)
%e A379734 (11,4,3,2,2)
%e A379734 (11,5,2,2,2)
%e A379734 (11,3,2,2,2,2)
%p A379734 b:= proc(n, i, t) option remember; `if`(n=0,
%p A379734 `if`(t=1, 1, 0), `if`(i<2, 0, b(n, i-1, t)+
%p A379734 `if`(i>n, 0, b(n-i, min(i, n-i), t/igcd(i, t)))))
%p A379734 end:
%p A379734 a:= n-> `if`(isprime(n), 1, b(n$3)):
%p A379734 seq(a(n), n=1..70); # _Alois P. Heinz_, Jan 07 2025
%t A379734 Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&Divisible[Times@@#,n]&]],{n,30}]
%Y A379734 These partitions are ranked by the odd terms of A326149.
%Y A379734 The strict version is A379735, allowing 1's A379733.
%Y A379734 A000041 counts integer partitions, strict A000009.
%Y A379734 A002865 counts partitions into parts > 1.
%Y A379734 A379666 counts partitions by sum and product, without 1's A379668.
%Y A379734 Counting and ranking multisets by comparing sum and product:
%Y A379734 - same: A001055, ranks A301987
%Y A379734 - divisible: A057567, ranks A326155
%Y A379734 - divisor: A057568, ranks A326149, see A379733
%Y A379734 - greater than: A096276 shifted right, ranks A325038
%Y A379734 - greater or equal: A096276, ranks A325044
%Y A379734 - less than: A114324, ranks A325037, see A318029, A379720
%Y A379734 - less or equal: A319005, ranks A379721, see A025147
%Y A379734 - different: A379736, ranks A379722, see A111133
%Y A379734 Cf. A069016, A318950, A319000, A319916, A324851, A325041, A326152, A379671, A379678.
%K A379734 nonn
%O A379734 1,4
%A A379734 _Gus Wiseman_, Jan 07 2025